Principal homogeneous space

A principal $ G $- object in the category of algebraic varieties or schemes. If $ S $ is a scheme and $ \Gamma $ is a group scheme over $ S $, then a principal $ G $- object in the category of schemes over $ \Gamma $ is said to be a principal homogeneous space. If $ S $ is the spectrum of a field $ k $( cf. Spectrum of a ring) and $ \Gamma $ is an algebraic $ k $- group (cf. Algebraic group), then a principal homogeneous space over $ \Gamma $ is an algebraic $ k $- variety $ V $ acted upon (from the left) by $ \Gamma $ such that if $ k $ is replaced by its separable algebraic closure $ \overline{k} $, then each point $ v \in V ( \overline{k} ) $ defines an isomorphic mapping $ g \rightarrow gv $ of the varieties $ V _{\overline{ {k}} } $ and $ \Gamma _{\overline{ {k}} } $. A principal homogeneous space $ V $ is trivial if and only if $ V(k) $ is non-empty. The set of classes of isomorphic principal homogeneous spaces over a smooth algebraic group $ \Gamma $ can be identified with the set of Galois cohomology $ H ^{1} (k,\ \Gamma ) $. In the general case the set of classes of principal homogeneous spaces over an $ S $- group scheme $ \Gamma $ coincides with the set of one-dimensional non-Abelian cohomology $ H ^{1} ( S _{T} ,\ \Gamma ) $. Here $ S _{T} $ is some Grothendieck topology on the scheme $ S $[2].

Principal homogeneous spaces have been computed in a number of cases. If $ k $ is a finite field, then each principal homogeneous space over a connected algebraic $ k $- group is trivial (Lang's theorem). This theorem also holds if $ k $ is a $ p $- adic number field and $ \Gamma $ is a simply-connected semi-simple group (Kneser's theorem). If $ \Gamma = \Gamma _{m,S} $ is a multiplicative $ S $- group scheme, then the set of classes of principal homogeneous spaces over $ \Gamma $ becomes identical with the Picard group $ \mathop{\rm Pic}\nolimits (S) $ of $ S $. In particular, if $ S $ is the spectrum of a field, this group is trivial. If $ \Gamma = \Gamma _{a,S} $ is an additive $ S $- group scheme, then the set of classes of principal homogeneous spaces over $ \Gamma $ becomes identical with the one-dimensional cohomology group $ H ^{1} (S,\ {\mathcal O} _{S} ) $ of the structure sheaf $ {\mathcal O} _{S} $ of $ S $. In particular, this set is trivial if $ S $ is an affine scheme. If $ k $ is a global field (i.e. an algebraic number field or a field of algebraic functions in one variable), then the study of the set of classes of principal homogeneous spaces over an algebraic $ k $- group $ \Gamma $ $\def\Sha{ {\mathop{\amalg\kern-0.30em\amalg}}}$ is based on the study of the Tate–Shafarevich set $ \Sha ( \Gamma ) $, which consists of the principal homogeneous spaces over $ \Gamma $ with rational points in all completions $ k _{V} $ with respect to the valuations of $ k $. If $ \Gamma $ is an Abelian group over the field $ k $, then the set of classes of principal homogeneous spaces over $ \Gamma $ forms a group (cf. Weil–Châtelet group).

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The notion of a principal homogeneous space is not restricted to algebraic geometry. For instance, it is defined in the category of $ G $- sets, where $ G $ is a group. Let $ G $ be a finite (profinite, etc.) group. Let $ E $ be a $ G $- set, i.e. a set $ E $ with an action $ G \times E \rightarrow E $ of $ G $ on it. Let $ \Gamma $ be a $ G $- group, i.e. a group object in the category of $ G $- sets, which means that $ \Gamma $ is a group and that the action of $ G $ on $ \Gamma $ is by group automorphisms of $ \Gamma $: $ (xy) ^ \gamma = x ^ \gamma y ^ \gamma $ for $ \gamma \in G $, $ x,\ y \in \Gamma $. One says that $ \Gamma $ operates compatibly with the $ G $- action from the left on $ E $ if there is a $ \Gamma $- action $ \Gamma \times E \rightarrow E $ on $ E $ such that $ ( \gamma x ) ^{g} = ( \gamma ^{g} )(x ^{g} ) $ for $ g \in G $, $ \gamma \in \Gamma $, $ x \in E $. A principal homogeneous space over $ \Gamma $ in this setting is a $ G $- set $ P $ on which $ \Gamma $ acts compatibly with the $ G $- action and such that for all $ x,\ y \in P $ there is a $ \gamma \in \Gamma $ such that $ y = \gamma x $. (This is the property to which the word "principal" refers; one also says that $ P $ is an affine space over $ \Gamma $.) In this case there is a natural bijective correspondence between $ H ^{1} (G,\ \Gamma ) $ and isomorphism classes of principal homogeneous spaces over $ \Gamma $ and, in fact, $ H ^{1} (G,\ \Gamma ) $( for non-Abelian $ \Gamma $) is sometimes defined this way.